Utility grids, such as electrical power grids, might have one or more generating plants that produce said utility for discharge by said grids. Said generating plants, however, may be subject to random adverse events that affect their ability to produce a demanded amount of said utility. Some utility grids have reservoirs for storing said utility for discharge to the plants to make up for the adverse events. These discharges of a utility from a reservoir to a plant due to random adverse events are referred to herein as “discharges”. The discharges from a reservoir to one of the plants might occur during a first time period. In order to make up for the discharges during a first time period, the plants connected to the reservoir by the grid may be required to make recharges of the utility to the reservoir during a subsequent second time period. These recharges are referred to herein as “recharges”. A technical problem to be solved, therefore, is controlling the recharges from the plants to the reservoir during the second time period so that in aggregate they make up for the discharges from the reservoir to the plants during the first time period.
One method controlling the recharges from each plant is to set them equal to the discharges from each plant. This is a problem, however, when the discharges are a small number of relatively large discharges such that any individual plant's discharge during a first time period would exceed said plant's ability to provide an equally large recharge during a second time period. Thus, a further technical problem to be solved is how to control the recharges from each of the plants connected to the reservoir by the grid so that in aggregate they make up for discharges that are large rare events.
Occasionally a discharge may initiate during a first time period but extend beyond the end of the first time period. These are referred to herein as long duration discharges. A further technical problem to be solved is determining how big a recharge should be for a second time period when a discharge has extended beyond the end of the first time period.